Question

Proof the relationship between marginal revenue and elasticity and show under what conditions marginal revenue will be positive, negative and zero?

Answer 1

Marginal revenue revenue is the change or additional total revenue generated when an extra product is produced and sold.

Elasticity is the Percentage change of one variable in response to changes in another.

The relationship between Marginal revenue and elasticity of demand can be illustrated as follows.

$R=P(Q)*Q,$

When we take the first order derivatives we get.

$(dR/dQ) =(dQ/dQ)*P +(dP/dQ)*Q$

$MR=dR/dQ=P+(dP/dQ)*Q$

$=P+(dP/dQ)*(Q/P)*P$

$=P*(1+1/e)$

Where R is the Total Revenue

PQ is the inverse demand function and $e<0$ is the price elasticity of demand.

When $MR>0$ any price cuts will increases total revenue and the demand is price elastic.

When $MR<0$ any price cuts will decrease total revenue and the demand is price inelastic

When $MR=0$ , the price elasticity of demand=1 signifying unitary elasticity. No changes in revenue.